What is most solved cells with "n" clues?

Everything about Sudoku that doesn't fit in one of the other sections

Postby coloin » Thu Feb 26, 2009 8:59 pm

JPFs
Code: Select all
+---+---+---+
|...|1..|...|
|.23|...|...|
|.4.|...|...|
+---+---+---+
|...|.4.|.2.|
|5..|...|...|
|1..|...|...|
+---+---+---+
|.6.|.2.|...|
|7..|...|1..|
|...|...|...|
+---+---+---+ 12 clues
actually
+---+---+---+
|.57|1..|...|
|.23|...|.1.|
|.41|...|...|
+---+---+---+
|3.6|.4.|.21|
|5..|.1.|...|
|1..|...|...|
+---+---+---+
|46.|.21|...|
|73.|...|1.2|
|21.|...|...|
+---+---+---+ 14 clues inserted
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Postby JPF » Fri Feb 27, 2009 12:06 am

Thanks coloin, I had noticed that.
My "quick solver" only looked at singles.

Here is a new improvement :

14/23
Code: Select all
 *-----------*
 |...|..1|...|
 |.2.|...|...|
 |...|...|...|
 |---+---+---|
 |3.1|...|...|
 |...|2..|..4|
 |..5|...|...|
 |---+---+---|
 |.6.|42.|...|
 |...|6..|5..|
 |...|...|13.|
 *-----------*

 *-----------*
 |.3.|..1|...|
 |.2.|3..|..1|
 |.1.|..2|..3|
 |---+---+---|
 |3.1|...|...|
 |...|25.|314|
 |2.5|13.|...|
 |---+---+---|
 |163|425|...|
 |...|613|542|
 |452|...|136|
 *-----------*

JPF
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Postby olimpia » Fri Feb 27, 2009 10:52 am

Cool. I like the 13/17 also.
My 13 clue board used 8 clue values and solve 16.
Your 13 clues uses only 6 values and solves 17. Values 1-6 expand all over while the lowly 7,8,9 are ignored and nothing gets known about them:(
Good work:!:
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Postby JPF » Sat Feb 28, 2009 7:14 pm

Here is an 11/10 ( 3 digits) :
Code: Select all
 *-----------*
 |...|...|...|
 |...|...|..1|
 |..2|..3|...|
 |---+---+---|
 |...|...|.1.|
 |...|..2|...|
 |.1.|...|..3|
 |---+---+---|
 |...|...|2..|
 |.3.|.1.|...|
 |2..|...|...|
 *-----------*     11 clues

 *-----------*
 |...|..1|..2|
 |...|.2.|..1|
 |1.2|..3|...|
 |---+---+---|
 |.2.|...|.1.|
 |...|1.2|...|
 |.1.|...|.23|
 |---+---+---|
 |..1|...|2..|
 |.3.|21.|...|
 |2..|...|1..|
 *-----------*     10 solve

more 11/10 :
Code: Select all
 *-----------*
 |...|...|...|
 |...|...|..1|
 |..2|..3|...|
 |---+---+---|
 |...|...|.1.|
 |...|..2|...|
 |.1.|...|..3|
 |---+---+---|
 |...|...|2..|
 |...|.3.|...|
 |2..|1..|...|     11/10
 *-----------*
 *-----------*
 |...|...|...|
 |...|...|..1|
 |..2|..3|...|
 |---+---+---|
 |...|...|.2.|
 |...|.1.|...|
 |.1.|.2.|...|
 |---+---+---|
 |...|3..|...|
 |1..|...|2..|
 |2..|...|...|     11/10
 *-----------*

eleven wrote:So to my surprise you need already 7 numbers for the 12/13 (at least my guess with 12+ was true)

Here a 12/14 with 6 digits :
Code: Select all
 *-----------*
 |...|...|...|
 |...|...|..1|
 |...|..2|...|
 |---+---+---|
 |...|...|...|
 |...|..3|.2.|
 |..1|...|...|
 |---+---+---|
 |...|14.|..5|
 |.6.|...|...|
 |23.|...|.4.|
 *-----------*    12 clues 
 *-----------*
 |...|..1|..2|
 |...|...|..1|
 |.1.|..2|...|
 |---+---+---|
 |...|...|.1.|
 |...|.13|.2.|
 |..1|...|...|
 |---+---+---|
 |...|146|235|
 |164|..5|...|
 |235|...|146|
 *-----------*    14 solve

JPF
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Postby eleven » Sat Feb 28, 2009 10:49 pm

Very good work again, JPF.

With your new puzzles the number of digits changes between 6 and 7 givens from 1 to 2, between 10 and 11 givens from 2 to 3 and between 12 and 13 from 3 to 6. This looks more smooth than before.

I tried to rearrange the 12 to 14 puzzles to show some similarities:
Code: Select all
 *-----------*    *-----------*
 |...|1..|...|    |...|1..|...|
 |..2|...|...|    |..2|...|1..|
 |...|...|...|    |..1|2..|...|
 +---+---|---|    +---+---|---|
 |...|...|..1|    |...|...|..1|
 |..3|..2|...|    |.13|..2|...|
 |...|...|...|    |...|..1|...|
 +---+---|---|    +---+---|---|
 |14.|5..|...|    |146|523|...|
 |...|...|6..|    |..5|...|614|
 |...|..4|32.|    |...|614|325|
 *-----------*    *-----------* 
 +---+---+---+    +---+---+---+
 |...|2..|...|    |...|2..|..4|
 |..4|...|3..|    |2.4|...|3..|
 |...|...|...|    |...|4..|2..|
 +---+---+---+    +---+---+---+
 |...|...|..2|    |...|3..|..2|
 |..3|..4|...|    |..3|.24|...|
 |...|...|...|    |..2|...|..3|
 +---+---+---+    +---+---+---+
 |12.|5..|...|    |126|543|...|
 |...|...|6..|    |..5|...|621|
 |...|1..|43.|    |...|162|435|
 +---+---+---+    +---+---+---+
 *-----------*    *-----------*
 |...|2..|...|    |3..|2..|..1|
 |..1|...|...|    |..1|3..|...|
 |...|...|...|    |..2|1..|..3|
 +---|---+---|    +---|---+---|
 |2..|...|..4|    |25.|...|314|
 |...|..5|...|    |13.|.25|...|
 |...|.31|...|    |...|.31|...|
 +---|---+---|    +---|---+---|
 |42.|6..|...|    |425|613|...|
 |6..|...|5..|    |613|...|542|
 |...|...|13.|    |...|542|136|
 *-----------*    *-----------*
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Postby JPF » Sun Mar 01, 2009 12:45 am

A new improvement 11/11 :

Code: Select all
 *-----------*
 |...|...|...|
 |...|...|..1|
 |..2|..3|...|
 |---+---+---|
 |...|...|.2.|
 |...|.1.|...|
 |.2.|...|...|
 |---+---+---|
 |...|..2|...|
 |..3|...|1..|
 |1..|...|..3|
 *-----------*    11 clues   

 *-----------*
 |...|..1|..2|
 |...|.2.|..1|
 |.12|..3|...|
 |---+---+---|
 |..1|...|.2.|
 |...|21.|...|
 |.2.|...|.1.|
 |---+---+---|
 |...|132|...|
 |2.3|...|1..|
 |1..|...|2.3|
 *-----------*    11 solve

JPF
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Postby eleven » Sun Mar 01, 2009 9:19 pm

Amazing again.

From this one you can drop the 3 in r8c3 to get a 10/10 with 3 digits:
Code: Select all
 *-----------*
 |...|...|...|
 |...|...|..1|
 |..2|..3|...|
 |---+---+---|
 |...|...|.2.|
 |...|.1.|...|
 |.2.|...|...|
 |---+---+---|
 |...|..2|...|
 |...|...|1..|
 |1..|...|..3|
 *-----------*
 *-----------*
 |...|..1|..2|
 |...|.2.|..1|
 |.12|..3|...|
 |---+---+---|
 |..1|...|.2.|
 |...|21.|...|
 |.2.|...|.1.|
 |---+---+---|
 |...|1.2|...|
 |2..|...|1..|
 |1..|...|2.3|
 *-----------*
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Posts: 1534
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Postby JPF » Mon Mar 02, 2009 2:17 am

well done eleven.
Here is the score now :
Code: Select all
  givens    qty solved        total    Digits                                                   
 ------    ----------        -----    ------                                                   
      4             1            5         1                                                   
      5             1            6         1                                                   
      6             3            9         1                                                   
      7             4           11         2                                                   
      8             5           13         2                                                   
      9             6           15         2                                                   
     10            10           20         3                                                   
     11            11           22         3                                                   
     12            14           26         6                                                   
     13            17           30         6                                                   
     14            23           37         6                                                   
     15            33           48         8                                                   
     16            47           63         7                                                   
     17            64           81         8                                                   

JPF
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Postby JPF » Tue Mar 03, 2009 10:44 pm

With n different digits, what is the minimum of clues required to place all those digits (9n) :

Here is a proposal :
Code: Select all
1 digit - 6 clues
 *-----------*       *-----------*
 |...|...|...|       |1..|...|...|
 |...|...|..1|       |...|...|..1|
 |...|..1|...|       |...|..1|...|
 |---+---+---|       |---+---+---|
 |...|...|...|       |...|1..|...|
 |...|...|.1.|       |...|...|.1.|
 |..1|...|...|       |..1|...|...|
 |---+---+---|       |---+---+---|
 |...|...|...|       |...|...|1..|
 |...|.1.|...|       |...|.1.|...|
 |.1.|...|...|       |.1.|...|...|
 *-----------*       *-----------*

2 digits - 10 clues
 *-----------*       *-----------*
 |...|...|...|       |...|..1|..2|
 |...|...|..1|       |2..|...|..1|
 |...|..2|...|       |1..|..2|...|
 |---+---+---|       |---+---+---|
 |...|...|.2.|       |...|...|12.|
 |...|.1.|...|       |...|21.|...|
 |.12|...|...|       |.12|...|...|
 |---+---+---|       |---+---+---|
 |...|...|2..|       |...|...|21.|
 |..1|...|...|       |..1|.2.|...|
 |.2.|1..|...|       |.2.|1..|...|
 *-----------*       *-----------*

3 digits - 13 clues
 *-----------*       *-----------*
 |...|...|...|       |..1|...|3.2|
 |...|...|..1|       |.23|...|..1|
 |...|123|...|       |...|123|...|
 |---+---+---|       |---+---+---|
 |...|..1|..3|       |...|..1|.23|
 |...|.3.|.1.|       |...|.32|.1.|
 |..2|...|...|       |132|...|...|
 |---+---+---|       |---+---+---|
 |...|2..|...|       |...|2..|13.|
 |.1.|...|...|       |21.|3..|...|
 |3..|...|2..|       |3..|.1.|2..|
 *-----------*       *-----------*

4 digits - 16 clues
 *-----------*       *-----------*
 |...|...|..1|       |...|3.2|.41|
 |...|...|.23|       |1.4|...|.23|
 |...|..4|...|       |23.|1.4|...|
 |---+---+---|       |---+---+---|
 |...|..1|4..|       |...|.21|43.|
 |..1|...|...|       |.41|..3|2..|
 |.23|...|...|       |.23|.4.|.1.|
 |---+---+---|       |---+---+---|
 |...|.1.|3..|       |...|41.|3.2|
 |...|2..|1..|       |3..|2..|1.4|
 |4.2|.3.|...|       |412|.3.|...|
 *-----------*       *-----------*

5 digits - 17 clues
 *-----------*       *-----------*
 |...|...|...|       |.12|5.3|..4|
 |...|..1|..2|       |5..|4.1|.32|
 |.34|...|.1.|       |.34|.2.|.15|
 |---+---+---|       |---+---+---|
 |...|...|34.|       |...|215|34.|
 |...|.3.|.5.|       |4.1|.3.|25.|
 |2..|...|...|       |253|..4|1..|
 |---+---+---|       |---+---+---|
 |...|.4.|...|       |...|14.|523|
 |.2.|...|..1|       |325|...|4.1|
 |1..|352|...|       |14.|352|...|
 *-----------*       *-----------*    5232 solutions

6 digits - 19 clues
 *-----------*       *-----------*
 |...|...|...|       |..1|.32|564|
 |...|..1|.23|       |465|..1|.23|
 |..2|45.|...|       |3.2|456|..1|
 |---+---+---|       |---+---+---|
 |...|...|45.|       |.23|1..|456|
 |...|6..|.1.|       |..4|6.5|312|
 |..6|2.3|...|       |156|243|...|
 |---+---+---|       |---+---+---|
 |.3.|...|...|       |63.|51.|24.|
 |.4.|.6.|...|       |24.|36.|1.5|
 |51.|...|.3.|       |51.|.24|63.|
 *-----------*       *-----------*    90 solutions

7 digits - 20 clues
 *-----------*       *-----------*
 |...|...|..1|       |.27|364|.51|
 |...|..2|.3.|       |61.|752|.34|
 |.45|...|6..|       |345|1..|672|
 |---+---+---|       |---+---+---|
 |...|...|7..|       |534|216|7..|
 |...|47.|5..|       |.62|47.|513|
 |..1|.3.|...|       |7.1|.35|246|
 |---+---+---|       |---+---+---|
 |...|5..|...|       |4..|521|367|
 |15.|...|42.|       |153|6.7|42.|
 |2.6|..3|...|       |276|.43|1.5|
 *-----------*       *-----------*    2 solutions

JPF
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Postby champagne » Tue Mar 03, 2009 11:20 pm

JPF wrote:With n different digits, what is the minimum of clues required to place all those digits (9n) :
JPF

Hi JPF,

This is not really a field for me, but if I got the point, we have plenty of examples in the 17 clues list showing that 17 is the minimum for 9 digits (no known puzzle with less than 17 clues).

Here the first puzzle in that list, solved by naked singles.


1= 000000010
2= 400000000
3= 020000000
4= 000050407
5= 008000300
6= 001090000
7= 300400200
8= 050100000
9= 000806000

champagne
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Postby JPF » Wed Mar 04, 2009 12:07 am

Obviously, when n = 8 or 9, the required number of clues is 17 :
Code: Select all
8 digits - 17 clues
 *-----------*       *-----------*
 |...|...|...|       |746|921|583|
 |...|...|.12|       |935|876|412|
 |...|.34|...|       |182|534|769|
 |---+---+---|       |---+---+---|
 |...|..5|3.6|       |894|215|376|
 |..7|...|...|       |567|493|128|
 |.21|.8.|...|       |321|687|954|
 |---+---+---|       |---+---+---|
 |.5.|1..|8..|       |253|169|847|
 |4..|...|...|       |478|352|691|
 |6..|7..|...|       |619|748|235|
 *-----------*       *-----------*    1 solution !

9 digits - 17 clues
 *-----------*       *-----------*
 |...|...|..1|       |273|964|851|
 |...|...|.2.|       |469|158|723|
 |...|..3|...|       |185|273|469|
 |---+---+---|       |---+---+---|
 |...|.4.|5..|       |821|346|597|
 |..6|...|3..|       |546|792|318|
 |..7|81.|...|       |397|815|246|
 |---+---+---|       |---+---+---|
 |.1.|.2.|..4|       |718|529|634|
 |.3.|...|.7.|       |632|48.|975|
 |95.|...|...|       |954|637|182|
 *-----------*       *-----------*    1 solution !

JPF
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Postby coloin » Wed Mar 04, 2009 12:32 am

I suspect JPF may well be trimming clues from the list of 17s !

"Obviously", in that if you can place all 8 digits - you can then place the remaining rookery. That almost certainly requires at least 17 clues.

I think you will find 7 digits - 16 clues - 2 solutions

C
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Posts: 1629
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Postby JPF » Wed Mar 04, 2009 2:36 am

coloin wrote:I suspect JPF may well be trimming clues from the list of 17s !

"Obviously", in that if you can place all 8 digits - you can then place the remaining rookery. That almost certainly requires at least 17 clues.
I don't understand.

coloin wrote:I think you will find 7 digits - 16 clues - 2 solutions
Indeed, but :
From the Gordon's list, all the 16 clues-7 digits-2 solutions are isomorphic to the puzzle already mentionned :

Code: Select all
 *-----------*       *-----------*
 |5.2|...|4..|       |562|3..|471|
 |...|71.|..3|       |.4.|716|253|
 |...|...|...|       |137|425|..6|
 |---+---+---|       |---+---+---|
 |...|..4|6..|       |3..|1.4|627|
 |.7.|2..|...|       |.7.|2.3|1..|
 |.1.|...|...|       |21.|..7|3..|
 |---+---+---|       |---+---+---|
 |6..|..2|...|       |6.1|.42|73.|
 |...|.3.|.1.|       |72.|63.|.14|
 |4..|...|...|       |4.3|.71|.62|
 *-----------*       *-----------*
7 digits, 16 clues, 2 solutions, but only 53 solvable cells (instead of 63)

Edit :
Not true as coloin pointed out below :
Code: Select all
 *-----------*
 |562|3..|471|
 |.4.|716|253|
 |137|425|..6|
 |---+---+---|
 |35.|1.4|627|
 |.74|263|1.5|
 |216|.57|34.|
 |---+---+---|
 |6.1|542|73.|
 |725|63.|.14|
 |4.3|.71|562|
 *-----------*
63 solvable cells

JPF
Last edited by JPF on Wed Mar 04, 2009 12:10 am, edited 1 time in total.
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Postby coloin » Wed Mar 04, 2009 3:03 am

Sorry, for being cryptic.

I assume you are taking 17 puzzles and removing clues and counting infered clues. Apologies if this is not the case. gsf is kindly working on this.

If you can palce all the 8X8 digits it means the 9th [rookery][ also is insertible.

Im glad you brought up the last point

The 16 clue pseudopuzzle with 2 solutions actually does "solve" to 63 clues.

r4c2 is 5 in both grid solutions

But i would describe the solving technique to be perhaps the equivalent of SE >7.2. [not sure what it actually is]

I dont know what the implications are of this in terms of the exercise - apart from in this case.

C
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Postby JPF » Wed Mar 04, 2009 4:23 am

coloin wrote:If you can palce all the 8X8 digits it means the 9th [rookery][ also is insertible.
Yes:)
coloin wrote:The 16 clue pseudopuzzle with 2 solutions actually does "solve" to 63 clues.
You are right.
Again, I forgot to turn off the button "quick" of my solver.
I edited my previous post.

The other values for different number of digits can probably be improved too.

JPF
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